Editing Current algebra (section)

==Example==
In a [[Non-abelian group|non-Abelian]] [[Yang–Mills]] symmetry, where {{mvar|V}} and {{mvar|A}}   are flavor-current and axial-current 0th components (charge densities),  respectively, the  paradigm of a current algebra is<ref>{{cite journal |last1=Gell-Mann |first1=M. |year=1964 |title=The Symmetry group of vector and axial vector currents |journal=Physics |volume=1 |issue=1 |page=63 |doi=10.1103/PhysicsPhysiqueFizika.1.63|pmid=17836376 |doi-access=free }}</ref><ref>{{harvnb|Treiman|Jackiw|Gross|1972}}</ref>
:<math> \bigl[\ V^a(\vec{x}),\ V^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ V^c(\vec{x})\ , </math> and 
:<math>
\bigl[\ V^a(\vec{x}),\ A^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x} - \vec{y})\ A^c(\vec{x})\ ,\qquad
\bigl[\ A^a(\vec{x}),\ A^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x} - \vec{y})\ V^c(\vec{x}) ~,</math>
where {{mvar|f}} are the structure constants of the [[Lie algebra]]. To get meaningful expressions, these must be [[normal order]]ed.

The algebra resolves to a direct sum of two algebras, {{mvar|L}} and {{mvar|R}}, upon defining
:<math> L^a(\vec{x})\equiv \tfrac{1}{2}\bigl(\ V^a(\vec{x}) - A^a(\vec{x})\ \bigr)\ , \qquad R^a(\vec{x}) \equiv  \tfrac{1}{2}\bigl(\ V^a(\vec{x}) + A^a(\vec{x})\ \bigr)\ ,</math>
whereupon
<math> \bigl[\ L^a(\vec{x}),\ L^b(\vec{y})\ \bigr]= i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ L^c(\vec{x})\ ,\quad
\bigl[\ L^a(\vec{x}),\ R^b(\vec{y})\ \bigr] = 0, \quad
\bigl[\ R^a(\vec{x}),\ R^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ R^c(\vec{x})~.
</math>